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The Use of High-Frequency Datain Financial Econometrics:Recent Developments
Peter Reinhard Hansen
Department of Economics, Stanford University
Stanford Conference in Quantitative Finance, 2010
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Financial Econometrics
The Society for Financial Econometrics (SoFiE) Founded 2008
Journal of Financial Econometrics (2003).
Oxford-Man Institute @ Oxford University (2007)
Volatility Institute @ NYU Stern (Robert Engle) (2009)
VLAB
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Introduction
Part 1: Accurate Measures of VolatilityRealized Measures of Volatility Computed from High-Frequency Data
Ideal Case: Realized VarianceNoisy Data (Market Microstructure)Empirical Properties of NoiseRobust Estimators:Realized Kernel & Markov Chain Estimator
Part 2: ApplicationsUtilizing Realized Measures for Volatility Modeling and Forecasting
Realized GARCH Models
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Realized Measures
of
Volatility
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High-Frequency Prices
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A Measure of Variation of Asset Prices
Suppose that Yt = logPt is a semi-martingale
Yt =
∫ t
0
audu +
∫ t
0
σudBu + Jt ,
where
a is a predictable locally bounded drift,σ is a cadlag volatility process,B is a standard Brownian motion, andJt =
∑Nt
i=1Di is a �nite activity jump process.
Quadratic Variation (over [0, 1]) is
QV =
∫1
0
σ2udu +
N1∑i=1
D2
i .
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Intraday Returns
Divide day into n subintervals
0 = T0 < T1 < · · · < Tn−1 < Tn = 1,
(e.g. equidistant Tj − Tj−1 = 1
n).
Intraday returns
yj ,n = YTj− YTj−1 , j = 1, . . . , n.
Daily return is the sum of intraday returns,
Y1 − Y0 = y1,n + · · ·+ yn,n.
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Realized Variance (Empirical QV)
Realized variance is de�ned by
RV =n∑
j=1
y2j ,n.
Properties (ideal case)
RVp→ QV as supi=1,...,n |Ti − Ti−1| → 0.
No jumps (i.e. Jt = 0) then
QV = IV :=
∫1
0
σ2udu, (integrated variance)
and √n(RV − IV )
d→ N(0,V ).
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Literature
Literature Invigorated byAndersen and Bollerslev (1998)Realized Variance useful for evaluation of GARCH models
Statistical Properties of Realized VarianceAndersen et al. (2001)Barndor�-Nielsen and Shephard (2002)Jacod (1994), Jacod and Protter (1998)
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The Great Tragedy of Science:
The Slaying of a Beautiful
Hypothesis by an Ugly FactThomas H. Huxley (1825-1895).
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The Great Tragedy of Science:
The Slaying of a Beautiful
Hypothesis by an Ugly FactThomas H. Huxley (1825-1895).
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High-Frequency Prices: AA 2007-05-04
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High-Frequency Prices: AA 2007-01-16
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High-Frequency Prices: AA 2007-01-26
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High-Frequency Prices: AA 2007-01-26 (zoom)
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High-Frequency Returns are Autocorrelated
Autocorrelation causes RV to be biased/inconsistent.
Computing RV with tick-by-tick returns says more about �noise� than�volatility�.
Ad-hoc resolution: Sample sparsely.
Compute RV using 5-minute returns.
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Sample Sparsely
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Volatility Signature Plot Reveals Bias Problem
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Price Without Noise
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Prices With Noise
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Properties of the Noise
Hansen and Lunde (2006)Journal of Business and Economic StatisticsInvited paper with Comments and Rejoinder.
Noise Ut is...
... serial dependent
... endogenous (not independent of Yt)
... has changed over time (tick size)
... is �small�.
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RobustRealized Measures
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Realized Kernel
Realized Autocovariances
γh =n∑
i=1
yi ,nyi−h,n.
So RV = γ0.
RK =∞∑
h=−∞k( h
H)γh,
where
k(·) is a kernel function andH is a bandwidth parameter.
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Kernel Functions
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Realized Kernel: Various Crises
Figure: Realized volatility for the period 1997-2009 (annualized) and the time ofsome of the major crises and events.
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Volatility Signature Plot
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Refresh Time and Semi-Stale Prices
Multivariate Problem
Asynchronous Trading (quoting)Refresh Time �aligns� observations,and induces additional noise
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Realized Beta from Realized Kernel
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Literature on Realized Kernel
Univariate Realized KernelBarndor�-Nielsen et al. (2008)
Multivariate Realized KernelBarndor�-Nielsen et al. (2010a)
Realized Kernels in PracticeBarndor�-Nielsen et al. (2009)
Relation to other estimatorsBarndor�-Nielsen et al. (2010b)
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Markov Chain Estimator(Personal Favorite)
Hansen and Horel (2009)
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Prices On A Grid
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The Markov Chain Estimator
Exploits discreteness of price changes.
Simple to compute
Estimate MC model (requires counting)Estimator computed with basic matrix operations
Inference...
Standard errors given in closed-formAsymptotics is reliable
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Empirical Example. GE 2004-11-01
x =
−3−2−11
2
3
P =
0.17 0.00 0.33 0.50 0.00 0.000.00 0.06 0.23 0.51 0.17 0.030.00 0.01 0.25 0.71 0.02 0.000.00 0.02 0.72 0.25 0.01 0.000.00 0.14 0.64 0.19 0.03 0.000.00 0.50 0.50 0.00 0.00 0.00
π =
0.000.020.480.480.020.00
.
Compute
Λπ = diag (π) , Z = (I − P + Π)−1 Π = ιπ′.
Markov Chain Estimator is: MC = x ′[
nclog
Λπ(2Z − I )]x
= x ′
0.007 −0.000 −0.001 0.000 −0.000 −0.000−0.000 0.035 −0.013 −0.003 0.010 0.002−0.000 −0.012 0.533 0.256 −0.004 −0.002−0.001 −0.004 0.250 0.532 −0.008 −0.000−0.000 0.008 0.004 −0.014 0.033 0.000−0.000 0.004 −0.001 −0.003 0.000 0.004
x = 0.7469.
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Filtering Argument
Observed Prices: XTj, j = 0, 1, . . . , n, with
∆XTj= XTj
− XTj−1 a homogeneous ergodic MC.
Consider �ltered pricesE(XTj+h
|FTj).
Easy to compute within the MC framework because
E(XTj+h|FTj
) = XTj+
h∑i=1
E(∆XTj+i|FTj
).
Even as h→∞!
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Martingale Representation
We haveXTj
= µj + YTj+ UTj
,
where
µj = µ · j , with µ = E(∆XTj);
YTj= lim
h→∞E(XTj+h
− µj+h|FTj)
is a Martingale; and
UTjis stationary, ergodic, bounded process.
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MC Estimator is RV of Filtered Price
XTj= µj + YTj
+ UTj,
MC estimator is basically the realized variance of YTj,
MC# = nx ′Λπ(2Z − I )x .
Consistent with a Gaussian limit distribution under appropriateassumption.
Log-correction
MC =nx ′Λπ(2Z − I )x
1
n
∑nj=1
X 2
Tj
.
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Con�dence Intervals
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Financial Crisis:
Markov Chain Estimates
USD/YEN
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Financial Crisis:
Markov Chain Estimates
SPY
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High-Frequency Data andForecasting
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Volatility Forecasting using High-Frequency Data
Hansen and Lunde (2011) (Oxford Handbook of Economic Forecasting).
HF data improves...
understanding of volatility dynamics � key for forecasting.understanding of the driving forces of volatility.For instance: Study of news announcements and their e�ect on the�nancial markets.evaluation of models/forecast
Realized measures...
good predictors of future volatility.led to new and better volatility models... yield more accurateforecasts.facilitate better estimation of complex volatility models.
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Realized GARCH Models
joint with
Albert Huang and Howard Shek
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ARCH and GARCH Models
ARCH-type models (Engle, 1982)
µt ≡ Et−1(rt) ht ≡ vart−1(rt).
Studentized returns
zt =rt − µt√
ht∼ (0, 1)
GARCH(1,1) (Bollerslev, 1986)
ht = ω + αr2t−1 + βht−1.
β ≈ 0.95 and α ' 0.05 in practice.
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GARCH Model
Squared returns, r2t−1, de�nes the dynamic of the conditional variance
ht = ω + αr2t−1 + βht−1.
r2t can be viewed as a noisy measure of volatility
Realized measures provide more accurate measurements of volatility.
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GARCH-X Model
Engle (2002), and many others
ht = ω + αr2t−1 + βht−1 + γxt−1.
xt is a realized measure of volatility (e.g. RV)
Huge improvement in the empirical �t.
Typically
γ ' 0.5.α ' 0. (ARCH parameter becomes insigni�cant)
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GARCH Model
Re-write GARCH(1,1) equation:
ht = ω + πht−1 + α(r2t−1 − ht−1)
π = α + β measures how persistent is volatility.
α ' �the strength of the signal r2t−1�
β ≈ 0.95 and α ' 0.05 in practice.
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GARCH is Slow
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GARCH-X with a Realized Measure is Fast
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GARCH-X is Incomplete
Data (rt , xt), but model only speci�es rt |rt−1,xt−1, . . .Simple case
rt =√htzt .
ht = ω + αr2t−1 + βht−1 + γxt−1.
Need a Model for xt .
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Realized GARCH
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Realized GARCH: Simple Case
GARCH-X structure
rt =√htzt
ht = ω + αr2t−1 + βht−1 + γxt−1
Measurement Equation completes the model
xt = ξ + ϕht + Errort .
xt is noisy measurement of QVt
QVt is ht + volatility shock.
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Logarithmic Speci�cation
Logarithmic speci�cation is preferred
log ht = ω + β log ht−1 + γ log xt−1.
log xt = ξ + ϕ log ht + τ(zt) + ut .
Leverage Function:
τ(z) = τ1z + τ2(z2 − 1)
Captures the joint dependence between
return shocks, ztvolatility shocks, τ(zt) + ut .
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Key Features of Realized GARCH
Empirical Features
Easy to estimate.Captures return-volatility dependence (leverage e�ect).Properties of multiperiod returns (skewness and kurtosis)Outperforms conventional GARCH
Theoretical Features (elegant mathematical structure)
Parsimonious
Tractable analysis (quasi maximum likelihood).Induced simple ARMA structure for both x and h
Natural extension of conventional GARCH
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Data
Dow Jones Industrial Average stocks and SPY (ETF).
2002-01-01 to 2007-12-31 as in-sample data and2008-01-01 to 2008-08-31 as out-of-sample.
For x , we use the realized kernel (RK) by BHLS (2008)
xt ≈ ht with open-to-close returnsxt<ht (on average) with close-to-close returns
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Linear Model (SPY Open-to-Close)
GARCH Equation
ht = 0.09(0.05)
+ 0.29(0.16)
ht−1 + 0.63(0.18)
xt−1
Measurement Equation
xt = −0.05(0.09)
+ 1.01(0.19)
ht +−0.02(0.02)
zt + 0.06(0.01)
(z2t − 1)︸ ︷︷ ︸τ(z)
+ ut
Standard deviation of ut : σu = 0.51(0.05)
.
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Linear Model (SPY Close-to-Close)
GARCH Equation
ht = 0.07(0.04)
+ 0.29(0.15)
ht−1 + 0.87(0.25)
xt−1
Measurement Equation
xt = +0.00(0.08)
+ 0.74(0.14)
ht +−0.07(0.02)
zt + 0.03(0.01)
(z2t − 1)︸ ︷︷ ︸τ(z)
+ ut
Standard deviation of ut : σu = 0.51(0.06)
.
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Log-Linear Model (SPY Open-to-Close)
GARCH Equation
log ht = 0.04(0.02)
+ 0.70(0.05)
log ht−1 + 0.45(0.04)
log xt−1 − 0.18(0.06)
log xt−2
Measurement Equation
log xt = −0.18(0.05)
+ 1.04(0.07)
log ht +−0.07(0.01)
zt + 0.07(0.01)
(z2t − 1)︸ ︷︷ ︸τ(z)
+ ut
Standard deviation of ut : σu = 0.38(0.08)
.
Persistence Parameter
π = β + (γ1 + γ2)ϕ = 0.986
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Log-Linear Model (SPY Close-to-Close)
GARCH Equation
log ht = 0.11(0.02)
+ 0.72(0.05)
log ht−1 + 0.48(0.06)
log xt−1 − 0.21(0.07)
log xt−2
Measurement Equation
log xt = −0.42(0.06)
+ 1.00(0.10)
log ht +−0.11(0.01)
zt + 0.04(0.01)
(z2t − 1)︸ ︷︷ ︸τ(z)
+ ut
Standard deviation of ut : σu = 0.38(0.08)
.
Persistence Parameter
π = β + (γ1 + γ2)ϕ = 0.987
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Estimated News Impact Curve
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Skewness and Kurtosis of Cumulative Returns
Figure: Skewness and kurtosis in line with empirical returns
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Multi-Period Forecast
Multi-period ahead predictions with the Realized GARCH model isstraightforward.
When p = q = 1. we obtain VARMA(1,1) structure[htxt
]=
[β γϕβ ϕγ
] [ht−1xt−1
]+
[ω
ξ + ϕω
]+
[0
τ(zt) + ut
],
so we can write, Yt = AYt−1 + µ+ εt .
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Realized GARCH Volatility
during the
Global Financial Crisis
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Realized GARCH: Global Financial Crisis
Figure: Conditional volatility during the global �nancial crisis with some of themajor events .Peter Reinhard Hansen (Stanford) Financial Econometrics November 2010 88 / 96
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Extensions
Realized GARCH
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Realized EGARCH with Multiple RM (w. Huang)
Multiple Realized Measures (like MEM by Engle & Gallo)
log ht = ω + β log ht−1 + γ ′ logXt−1 + τ(zt−1)
logXt = ξ +ϕ log ht + δ(zt) + Ut .
RK crowds out RV and Range (High minus Low)
var(Ut) yields information about accuracy about RMs.
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Multivariate Realized GARCH (w. Lunde & Voev)
Build Realized GARCH for Market Returns, r0,t = µ0 +√h0,tz0,t .
Asset i 's returns, ri ,t = µi +√hi ,tzi ,t , conditional on r0,t , x0,t .
Key in this model:ρt = cov(z0,t , zi,t |Ft−1).
Measurement equation with Fisher transform
z(ρt) = a0i + b0iz(ρt−1) + c0iz(y1,t−1).
1-factor structure where we can extract
βt = ρt
√hi ,t/h0,t .
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Correlation Measurement Equation (CVX)
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Time Series for ρt and βt (CVX)
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Cross Sectional Beta-Quantiles
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Conclusion
Realized Measures
Empirical Issues with High-Frequency DataRealized Variance, Realized Kernel, Markov Chain Estimator
Volatility Forecasting
Realized GARCH
Easy to estimate, Tractable, can explain empirical �stylized facts�.Multivariate extension very promising.
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